1 Introduction


Quantum Black Hole Entropy, Localization

and the Stringy Exclusion Principle


João Gomes


Institute of Physics, University of Amsterdam, Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands Institute for Theoretical Physics, University of Utrecht, Princetonplein 3584 CC Utrecht, The Netherlands j.m.vieiragomes@uva.nl


ABSTRACT

Supersymmetric localization has lead to remarkable progress in computing quantum corrections to BPS black hole entropy. The program has been successful especially for computing perturbative corrections to the Bekenstein-Hawking area formula. In this work, we consider non-perturbative corrections related to polar states in the Rademacher expansion, which describes the entropy in the microcanonical ensemble. We propose that these non-perturbative effects can be identified with a new family of saddles in the localization of the quantum entropy path integral. We argue that these saddles, which are euclidean geometries, arise after turning on singular fluxes in M-theory on a Calabi-Yau. They cease to exist after a certain amount of flux, resulting in a finite number of geometries; the bound on that number is in precise agreement with the stringy exclusion principle. Localization of supergravity on these backgrounds gives rise to a finite tail of Bessel functions in agreement with the Rademacher expansion. As a check of our proposal, we test our results against well known microscopic formulas for one-eighth and one-quarter BPS black holes in and string theory respectively, finding agreement. Our method breaks down precisely when mock-modular effects are expected in the entropy of one-quarter BPS dyons and we comment upon this. Furthermore, we mention possible applications of these results, including an exact formula for the entropy of four dimensional black holes.

April 13, 2018

1 Introduction

Supersymmetric localization [1, 2] has lead to the possibility of evaluating exactly the path integral that computes the quantum entropy [3] of BPS black holes. This technique has been particularly successful for computing perturbative quantum corrections to the Bekenstein-Hawking entropy in toroidal compactifications , where an almost exact matching with the microscopic theory was obtained [4].

The goal of this work is to address instead non-perturbative corrections to black hole entropy related to polar states in the microscopic theory. We want to understand the origin of these effects, perhaps as new saddle points of the path integral, and if so how to compute quantum corrections around each saddle. In toroidal compactifications, such non-perturbative effects are not present, which in a way is what explains the simplicity of the microscopic formulas. Nevertheless, for and compactifications, these non-perturbative effects are crucial to understand black hole entropy in the limit of very large central charge, which is where the four dimensional semiclassical description holds. Though exponentially subleading, these non-perturbative effects can become relevant when the number of polar states grows exponentially, which is the case of large central charge.

Recent attempts to compute exactly the quantum entropy, rely on the four dimensional effective action, which includes instanton effects in the prepotential of supergravity. In order to address the problem of non-perturbative corrections within the context of supersymmetric localization, we need to first understand the UV dynamics that are responsible for those effects. With this in mind, the following question arises: can we rely on effective field theory such as supergravity or do we need the full string theory?

Another issue is of concern. Since localization is an off-shell computation and as such does not depend on the values of the coupling constants, it is valid at strong and weak coupling. Translating to supersymmetric black holes and their quantum entropy, this means that the localization computation should hold for any value of charges. The reason is that, in string theory, the scalar fields, which play the role of the coupling constants, become functions solely of the charges at the black hole attractor. However, in view of the AdS/CFT correspondence and the fact that we are working in the microcanonical ensemble, this raises many issues related to the validity of effective field theory. For example, the characteristic length scale of the geometry is a function of the charges, and so by scaling these it is possible that a particular dimensional point of view is more appropriate than other. Conversely, we may ask which microscopic Lagrangian are we localizing?

To better understand these issues, we take a pedestrian approach. We start by recalling the original localization computation of [2] in four dimensional supergravity and discuss its validity using effective field theory. Then in section §1.2 we consider the five dimensional point of view. We argue this is more appropriate to describe the physics of the Rademacher expansion. In section §1.3, we discuss the connection between non-perturbative effects in the black hole entropy and the counting of polar states. Along the way, we present the main lines of our solution, which, in essence, is a reformulation of the OSV formula. We require this formula to be compatible with the Rademacher expansion and to reproduce at the same time the counting of polar states.

1.1 The four dimensional point of view and the OSV formula

In [2], it is shown that the path integral of supergravity 1on , which computes the entropy of a BPS black hole, reduces to a finite dimensional integral by means of supersymmetric localization. The answer for the black hole degeneracy is schematically of the form

(1.1)

where is the four dimensional holomorphic prepotential that encodes the couplings of the vectors to the Weyl multiplet, and are respectively the electric and magnetic charges; the integration variables correspond to normalizable modes that are left unfixed by localization.

The result (1.1) is a reincarnation of the conjectured OSV formula [5], which relates the black hole quantum entropy to the topological string partition function. In the original formulation [5], the reason this happens is because the supergravity prepotential in (1.1) is directly related to the perturbative free energy of the topological string on the Calabi-Yau compactification manifold. To be more precise, the topological string computes four dimensional higher derivative terms, also called F-type terms, of the form () with being the anti-self-dual curvature two-form and the graviphoton field [6, 7]. The 2 are defined in a perturbative expansion in powers of the topological string coupling constant , that is,

(1.2)

For , the tree level term can be approximated by with the intersection matrix of the Calabi-Yau threefold, while the function is the one-loop correction which approximates to . The corrections of order are due to worldsheet instantons, while the parameter can be identified with the second Chern-class (tangent bundle) of the Calabi-Yau. The map between the topological string variables and the supergravity fields is the following: the complexified Kähler parameter and , where is the dilaton that sits in the supergravity multiplet and are the vectormultiplet complex scalar fields. 3.

For a general Calabi-Yau, the function (1.2) is known only as an asymptotic expansion in . If one tries to use localization at the level of the four dimensional effective Lagrangian as in [2] we run into serious problems; not only we have to constrain the scalar to very large values, but we also need to include an arbitrary number of higher derivative corrections. The best we can do is to compute order by order in the inverse of the charges using perturbative methods.

Nonetheless, in compactifications that preserve more supersymmetry, like in and , the prepotential (1.2) is one-loop exact, that is, all vanish. In this case, the tree level free energy is given exactly by and the one-loop contribution , with being a worldsheet instanton partition function. Since the prepotential is now one-loop exact one might expect to be able to use supersymmetric localization at the level of the four dimensional effective action. As a matter of fact, previous work shows that in the theory it is possible to reproduce exactly all the perturbative corrections to the area formula [4] including non-perturbative corrections, related to orbifold geometries, that depend on intricate Kloosterman sums [8]. In the case, however, the situation is not so satisfactory because the localization integral (1.1) is not able to reproduce all the features predicted by the microscopic theory. In particular, it fails to reproduce the measure that is known from microscopics even after taking into account the one-loop determinants [9, 10]. In a way, this is partially justified from the microscopic studies [11, 12, 13]. These studies predict a measure to (1.1) that depends strongly on the worldsheet instantons. The precise dependence is of the form

(1.3)

where can be identified with a worldsheet instanton partition function [14] that appears in topological string free energy, and is a quadratic magnetic charge invariant. Since the instanton corrections carry non-trivial information about the Calabi-Yau manifold, related to Gromov-Witten invariants, it would be puzzling if the four dimensional localization computation, including the one-loop determinants, could explain those corrections. In general the determinants are given in terms of equivariant indices of the four dimensional background with no connection to the Calabi-Yau invariants. Instead, one needs to understand the dynamics that are responsible for the quantum corrections that one observes at the level of the microscopic measure.

The near-horizon geometry can help clarify some of these issues by drawing a clear contrast between four and five dimensional physics. Lets consider the attractor geometry of the black hole in IIA and uplift to M-theory. The near horizon geometry [15] is spherically symmetric and contains a local factor which consists of the M-theory circle fibered over , that is,

(1.4)

Here parametrizes the circle, is the Kaluza-Klein gauge field, is the physical size which can be taken to be large, and is the Calabi-Yau metric. Both and factors inside the brackets have unit size in string units. When reducing to four dimensional IIA string theory the radius of the circle becomes the scalar in (1.2). Given the map between the supergravity varibles and the topological string, finite radius means finite topological string coupling constant. So for finite radius, the Kaluza-Klein modes, that one obtains after compactification on the circle, have masses which are comparable to the inverse size and thus the solution is best described in five dimensional supergravity. In contrast to four dimensions, the part of five dimensional Lagrangian that contains the couplings of the vectors to the Weyl multiplet, is completely determined by the parameters and , which appear in the topological string free energy. Therefore, at the level of the Lagrangian that one obtains after dimensional reduction on the Calabi-Yau, there seems to be no information about the worldsheet instantons.

1.2 The five dimensional point of view and the Rademacher expansion

The four dimensional problem just described holds in the regime for which the radius of the circle is parametrically smaller than the size of , or equivalently in the regime of weak topological string coupling constant. However, if supersymmetric localization should hold for any value of the charges 4, the regime of 5 should be equally well valid, but this corresponds to take the five dimensional point of view. In the following, we shall argue that the microscopic Rademacher expansion is more appropriate to describe the five dimensional physics, and we build our solution based on this idea.

As we explain later in more detail, localization at the level of the five dimensional theory, initiated in [12, 16], gives instead the finite dimensional integral

(1.5)

with the ”classical” prepotential that we define as

(1.6)

This is the one-loop approximation of the prepotential (1.2) without the worldsheet instanton corrections. It is thus clear that the dependence on the worldsheet corrections (1.3) must arise from other effects. In contrast, the localization integral captures only perturbative quantum corrections around the attractor background (1.4), as an expansion in the area.

We can check that the integral (1.5) matches the expectations from the microscopic theory. Performing the various integrals, it is found that the final result matches the microscopic degeneracy - a Bessel function, valid precisely for large [12], including the measure factor in as well as in all CHL examples in both and compactifications. For this reason, we consider the five dimensional point of view to be a step closer to understanding the quantum measure and the role of the worldsheet instantons.

Besides the leading Bessel, the microscopic answer contains a series of subleading Bessel contributions. Schematically, they arise after expanding the functions (1.3) as instanton sums and then performing appropriate integrals [12, 13]. As a result, non-perturbative corrections to black hole entropy are generated. Remarkably, this series of Bessel contributions can be matched, to a certain extent, to the polar state contributions in a mock Jacobi Rademacher expansion [13].

The main goal of this work is to clarify the origin of the subleading Bessel functions, in a way consistent with the quantum entropy functional. Even though they are non-perturbative for large , they can become relevant in the opposite regime of , which occurs for large central charge, because the number of Bessel functions can increase exponentially. According to the supergravity/topological string map, that regime corresponds to the four dimensional point of view, which is why it is crucial to understand the origin of the subleading Bessels. Preliminary steps in this direction were already taken in [12], where it was suggested that the subleading Bessel contributions could arise from additional configurations of the full string theory path integral. To better understand the claim lets look in more detail to the Rademacher expansion [17], which is an exact formula for the Fourier coefficients of modular forms- the black hole microscopic degeneracy 6. Schematically one has

(1.7)

where is the black hole degeneracy, which is a function of the charge combination , is the degeneracy associated with the polar terms and are Kloosterman sums; each of the integrals are modified Bessel functions of the first kind. The microscopic answer derived in [13] has precisely this form after neglecting the terms. Also, in this work we will only be considering the terms with , usually referred as polar Bessels. For with fixed, the Bessel functions have saddle points at

(1.8)

and growth

(1.9)

The leading contribution in (1.7) therefore comes from the term with maximal , with the polarity. In terms of the bulk physics, this is precisely the leading Bessel function that one obtains by evaluating the localization integral (1.5), with given by the charge combination . The terms with generate exponentially suppressed corrections and hence are non-perturbative. Furthermore, the value of can be identified with the topological string coupling constant , and so we see that the saddles (1.8) lie at large values of , when the five dimensional point of view makes sense. In this regime of charges it is thus pertinent to ask to what the non-perturbative saddles correspond from the five dimensional point of view. This is puzzling because, given the localization computation (1.5), there seems to be no room for any other additional contribution to the path integral. Though, it is possible that these saddles arise from other configurations in the full M-theory path integral. From which ones and how do they contribute? These are some of the questions that we want to address.

Our approach is mainly heuristic. In essence, we propose that the full path integral of M-theory receives the contribution of a new family of configurations which are euclidean geometries of the type . The factor is a local geometry, such as (1.4), with euclidean time contractible, and guarantees that after reduction on the circle, one obtains the four dimensional attractor background. This also follows from the fact that the four dimensional localization equations fix the geometry to be exactly [19]. Therefore we see that from a five dimensional point of view there is not much room for the space of allowed geometries, except that it must have a circle fibered over the attractor geometry.

To be consistent with the path integral and the localization computation, we argue that the new configurations are exact solutions of different five dimensional Lagrangians that we see as effective descriptions. The difference between Lagrangians is a finite renormalization of the parameters that define the theory such as (1.6), which is the gauge-gravitational Chern-Simons coupling in five dimensions. The supersymmetric localization computation at the level of the five dimensional theory reproduces the tail of polar Bessel functions observed in the microscopic answers, including the exact spectrum of . That is, for each euclidean geometry we find a Bessel function with index and argument given as in (1.7), thus explaining the origin of the non-perturbative effects from a five dimensional point of view.

The renormalization of has an additional effect. It corrects the physical size of the geometry in such a way that it can become zero, thus imposing a physical condition on the number of geometries. We find that this bound is in perfect agreement with the stringy exclusion principle [20]. The bound on the number of possible geometries is essentially the reason why there is only a finite number of polar Bessel functions in the Rademacher expansion. In the semiclassical limit, that is, when the central charge is very large, the number of geometries close to maximal polarity is dense which allows for a saddle point approximation. The result of this can be identified with the dilute gas approximation of the path integral as in [21], and the non-perturbative corrections around that saddle correspond to excitations of the fields dual to the chiral primary states.

It is also instructive to compare the above proposal with the effective field theory computation in , which is the setup considered in [6, 7] and revisited in [22], for deriving the Gopakumar-Vafa formula. They consider a one-loop computation for the Kaluza-Klein modes of vectors and hypermultiplets on the circle , in the background of a self-dual graviphoton field. The result of this computation is the four dimensional higher derivative terms proportional to the topological string free energies (1.2). At the on-shell level we do not expect the and computations to differ much when the curvatures are very small. So computing the instanton contributions, in , to the on-shell entropy function, we can generate non-perturbative corrections to the entropy area formula [13]. Nevertheless, at the quantum level, placing the theory on the background (1.4), leads to problems related to the stringy exclusion principle [20]. The path integral of the reduced theory 7 on 8 contains fluctuations that are not unitary and hence are expected to backreact on the background solution [23]. The role of the exclusion principle is to artificially truncate the perturbative spectrum of fluctuations in agreement with the dual field theory. The exclusion principle is more relevant for small central charge which makes it a non-perturbative effect. The way we circumvent this problem is by considering the full M-theory path integral, instead of using the effective five dimensional Lagrangian with the massive hypermultiplets that are needed to obtain the Gopakumar-Vafa formula.

In fact, we show that in the limit of charges for which the circle becomes parametrically smaller than the size of , while keeping the curvature small, we recover the perturbative partition function, in an expansion in the charges, as determined by the four dimensional effective action. This regime of charges is obtained by scaling faster than such that we have at the saddle point. In this regime, we shall recover the Gopakumar-Vafa corrections to black hole entropy. We explain, in addition, how the on-shell logarithmic corrections computed in [24, 25] arise from our formalism.

To put it more explicitly, we provide with a non-perturbative formula for black hole entropy that correctly interpolates between the five and four dimensional physics. For small central charge , one has only a small number of geometries and thus also a small number of Bessels. Schematically we have the gravitational answer

(1.10)

which is the Bessel function, in agreement with the Rademacher expansion. Whereas for large central charge the high density of geometries, and so of Bessels, allows for a saddle point approximation. For the and models, we recover partially the OSV formula, that is, the holomorphic part, with corrections that we can systematically compute,

(1.11)

Here , which encodes the holomorphic free energies, can be seen as a canonical partition function for the non-perturbative effects. These effects can then be related to the Gromov-Witten worldsheet instantons.

1.3 The polar state side of the story

So far we have described the problem from the black hole point of view. However, there is another side to this story, which is not directly connected to black holes. This is the context of polar states and its relation to chiral primary states. We will study these states, which can be seen as bound states, and we shall argue that the proposed geometries are the bulk duals of these microscopic configurations, after a modular transformation.

Polar states are characterized by having negative charge discrimant in the R sector of the CFT. Since black holes have necessarily positive charge discriminant, polar states must correspond in the bulk to configurations that do not form single center black holes. However, the reason why the information about polar states enters in the black hole counting formula (1.7) is due to the modular properties of the CFT partition function. In fact, knowing the spectrum of polar states defines completely the spectrum of non-polar states, and so using modularity we can relate one to the other.

There is an extensive literature on the problem of determining the spectrum of polar states and then use modularity to study corrections to black hole entropy [21, 26, 27]. One of the most complete of such studies is the work of Denef and Moore. Succintly, they perform an extensive study of polar multi-center black hole solutions in four dimensional supergravity, with the goal of determining their contribution to the spacetime index. The main ingredients used are attractor flow trees [28] and the wall-crossing phenomena. They find that at large topological string coupling, the main contribution to the index comes from two center black hole solutions, corresponding to a configuration of a single and a single anti- () with worldvolume fluxes, located at different positions in . The fluxes considered contain, besides the smooth part, a singular component, which is represented by ideal sheaves. Their contribution to the index gives rise to a refined version of the OSV answer, which includes a measure of the sort described by (1.3).

The multi-center black hole solutions studied by Denef and Moore, admit a decoupling limit after an uplift to five dimensions [29, 30]. In particular for the two center solution, the region near the core, where the and sit, is zoomed in, and the decoupled geometry becomes asymptotically with no black holes inside [30]. It can be shown that these solutions carry Virasoro charges consistent with the values expected for . Nevertheless, this result holds only for very close to its maximal value. We revisit this construction and establish a parallelism with our solutions.

In all the works on black hole entropy through polar state counting, one uses the as an intermediate step. First, we build a partition function for the polar states , with the complex structure of the torus where the lives, and are chemical potentials. Then, we use modularity to construct the black hole partition function [31] as

(1.12)

This is only an approximate equality because we are not including the contribution due to other elements in the modular group. Nevertheless, for the purpose of studying non-perturbative effects due to the polar contributions, it is enough to consider only the modular transformation .

From the CFT point of view, naturally receives the contribution from only a finite number of states, those with negative discriminant. Nevertheless, from the bulk, one has to truncate artificially the perturbative spectrum of Kaluza-Klein fields on , which are the fields dual to chiral primary states (polar states in the R sector). The truncation is known as the stringy exclusion principle [20] and asserts that quantum gravity in is inherently non-perturbative.

The solution that we propose in this work is greatly inspired by the polar geometries studied in [30]. The asymptotically polar configurations have a complicated geometry, but for large central charge we can write the metric as a background global geometry plus corrections proportional to the singular fluxes, which are of the order of the inverse of central charge. A modular transformation makes the euclidean time circle contractible giving rise to asymptotically black hole like geometries [20, 32, 33]. However, in view of the localization computation that we want to perform, these solutions are not satisfactory because they do not have an exact factor [19, 16]. In a sense, which we would like to understand in more detail, our solutions are the non-perturbative analog, when taking into account the full string theory, of these modular transformed polar configurations. Conversely, we expect the fully quantum corrected polar configuration to have an exact global factor.

To build intuition about the quantum corrected polar configurations we proceed as follows. The approach described in [27, 30] considers first the backreaction of a two center configuration in four dimensions and then its uplift to M-theory. Equivalently, we can think of the same bound state as a brane configuration with worldvolume fluxes . It is well known that such configuration uplifts in M-theory to a Taub-Nut/anti-Taub-Nut geometry with fluxes , with the field strength of the M-theory three-form and is a normalizable two form of the Taub-Nut geometry. Therefore, fluxes on the branes map to fluxes in M-theory. If the worldvolume fluxes are ideal sheaves [27] we can generate arbitrary charges while keeping fixed the charge. We argue that the presence of such fluxes on the Calabi-Yau can induce corrections on the five dimensional Lagrangian after reduction. Then, solving the full five dimensional equations of motion we find instead the black hole geometry without corrections, but with the physical size (1.4), and the attractor values of the scalar fields depending explicitly on the fluxes. Localization on these backgrounds reproduces the finite tail of polar Bessel functions in the Rademacher expansion, thus setting the stage for a possible derivation of the solutions that we propose. The presence of singular M-theory fluxes can be understood as quantum fluctuations of the Kähler class of the Calabi-Yau, which allows us to make a connection with the quantum foam picture of topological strings studied in [34].

To guide the construction of our solution, we will revisit the counting of chiral primary states on and its relation to (1.12) following [21, 26]. Since our goal is to interpret the quantum black hole entropy as a partition function of M-theory, we will want to reproduce the counting of chiral primaries purely in terms of the eleven dimensional M-theory fields, and this will lead us inevitably to the polar configurations with singular fluxes. The counting consists essentially in building multi-particle states on top of the vacuum by acting with the quanta that we obtain from the fields dual to the chiral primary states [35, 36]. To do that we need to analyze the Kaluza-Klein tower of fields on coming from the supergravity fields and the , and (anti)- branes, wrapping cycles on the Calabi-Yau. Contrary to [21], which works in the dilute gas approximation, we will reconsider the same counting but for finite central charge. Imposing the stringy exclusion principle and spectral flow symmetry will enable us to reproduce the non-perturbative corrections induced by the polar Bessels in the Rademacher expansion, including the polar coefficients .

1.4 Outline

The plan of the paper is as follows. In section §2, we start by describing in more detail the Rademacher expansion and connect it to previous work on black hole entropy and localization in supergravity. Then we review the microscopic formula for the entropy of one-quarter BPS black holes, which includes the effect of the subleading Bessel contributions. We use this formula as a check of our proposal and later make comments on black hole entropy. Before moving to the discussion about the configurations, in section §3 we review the problem of counting chiral primaries on , which includes M2 and anti-M2-branes wrapping holomorphic cycles of the Calabi-Yau [20, 21]. Taking into account the stringy exclusion principle and spectral flow symmetry we obtain a formula that agrees precisely with the microscopic answer at finite charges; this formula will serve as a guide for the solution that we propose. Then in section §4, we review the configurations with worldvolume fluxes and their decoupling limit. We argue for the existence of a family of configurations and then in section §5 we compute the partition function using localization. The result of this is a finite sum over Bessel functions, whose spectrum is in agreement with the spectrum of polar states of a Jacobi form. Finally in section §6 we discuss a connection between our solutions and the quantum foam picture of non-perturbative topological string.

2 Quantum saddle points from Rademacher expansion

The Fourier coefficients of Jacobi forms of non-positive weight admit an exact expansion in terms of an infinite sum of Bessel functions. This expansion is known as Rademacher expansion [32] and provides with a simple way to address the asymptotic behaviour of the integer Fourier coefficients. We review this expansion and connect to previous work on black hole entropy corrections.

Consider a Jacobi form of level and negative weight , with Fourier expansion

(2.1)

The coefficients with non-negative discriminant , which are known as non-polar coefficients, admit an exact expansion in terms of an infinite sum of Bessel functions. Known as Rademacher expansion [37, 32] it has the form

(2.2)

The coefficients have negative discriminant, or polarity, , and are thus the polar coefficients, and are Kloosterman sums [38]. The structure in (2.2) is completely fixed by modularity except for the knowledge of the polar coefficients.

One of the great advantages of the expansion (2.2), is that it is very appropriate to the study of asymptotics. For with finite each of the Bessel functions have a saddle point at

(2.3)

Around each saddle we can expand perturbatively in powers of such that

(2.4)

where the denote perturbative corrections in powers of around each of the saddles , we are ignoring a normalization factor for each of the perturbative series. Therefore we see that the sum of polar contributions results in a tail of exponentially suppressed terms relative to the term of maximal polarity.

For holomorphic Jacobi forms9, the leading term in the expansion (2.4) comes from the most polar term, which has . From the bulk physics point of view, we can identify the leading exponential growth with the black hole entropy area formula, since we have , where is the area of the black hole horizon. Similarly, in the near-horizon attractor geometry (1.4) the saddle value of is identified with .

In addition, we can compute quantum perturbative corrections to the leading saddle using localization and a connection to Chern-Simons theory [12]. The result is the finite dimensional integral (1.5), which we review in section §5.1 using localization at the level of five dimensional supergravity. The idea of [12] is roughly the following. We start with the four dimensional localization integral (1.1) and approximate the prepotential according to the regime , where the leading saddle lives. Indeed, the on-shell complexified Kähler class becomes large, that is, and the one-loop topological string free energy approximates to leading to the classical prepotential (1.6). The quantum measure, on the other hand, is fixed by a zero mode argument using the Chern-Simons formulation. Though this formulation is well justified in the regime of , it is argued in [12] that the zero mode argument can be extrapolated also for the regime , which allows to define a quantum measure.

The subleading saddle points in (2.4), corresponding to the polar terms with , lead to exponentially suppressed corrections of the form

(2.5)

with . Given what we know already for the leading Bessel function in terms of bulk physics, it becomes pertinent to understand what is the origin of the subleading saddles from the quantum entropy functional. In fact, there is partial understanding for the leading saddles in the tails (2.4),

(2.6)

at the level of the quantum entropy path integral [39, 8]. In this case, the subleading terms that grow as arise after including in the path integral orbifolds of locally geometries [33]. The orbifold explains the exponential growth that characterizes them, because the area is reduced by a factor of due to the orbifold.

There is something particular to the subleading polar terms when compared with the orbifold saddles, which is partially the reason why their bulk interpretation is more difficult. While for the orbifold saddles the values of are consistent with the attractor background, for the subleading polar saddles (2.5) the values of (2.3) are quite distinct from the on-shell attractor background values, which can be determined from the leading Bessel; they differ from finite renormalizations. If these saddles indeed correspond to bulk saddle configurations, then they can not be solutions of five dimensional supergravity that one obtains after compactification on the Calabi-Yau.

2.1 Degeneracy from Siegel Modular Forms

In the following we present a study of the microscopic degeneracy for dyons in and CHL orbifold compactifications [12, 13]; we describe in detail the role of the polar contributions. Though, our considerations are valid also for compactifications, we will use the answer as a check of our proposal.

The index of -BPS dyons in four dimensional compactifications, has generating function the reciprocal of a Siegel modular form , that is,

(2.7)

Here is the weight of the modular form under a congruence subgroup of , and depends on the orbifold compactification. The integers label respectively the T-duality invariants , and with electric charges and magnetic charges (in a particular heterotic frame). For further details we point the reader to [40].

Conversely we can extract the integers - the black hole degeneracies, by performing an inverse Fourier transform. The function contains poles, and thus by deforming the contour, the integral picks the residues at those poles. It turns out that the dominant contribution to the black hole degeneracy is

(2.8)

with

(2.9)

The functions are modular forms of weight , with Fourier expansion

(2.10)

Choosing appropriately the contour in (2.8) [12], we can rewrite the degeneracy (2.8) as a finite sum of integrals of Bessel type, that is,

(2.11)

with

(2.12)

and

(2.13)

with for and CHL models respectively. Integrating over we obtain a sum over Bessel functions with the series resembling the Rademacher expansion (2.2). This has led the authors in [13] to test this possibility against an exact mock-Jacobi Rademacher expansion [18].

For , extremization of (2.12) gives the Cardy formula

(2.14)

where can be identified with the left central charge (of the non-supersymmetric side of the ). The values of and at the saddle point are

(2.15)

From the bulk physics, and are mapped respectively to the values of the scalar fields and of the four dimensional supergravity.

For we can proceed similarly. Each term has exponential growth

(2.16)

and the values of the saddles are

(2.17)

From here we see that these saddles differ from (2.15) by finite renormalizations parametrized by .

3 Polar states from M2 and anti-M2 branes

In this section, we review the problem of counting chiral primary states from the bulk theory on , with a Calabi-Yau manifold, and how it connects to the study of black hole entropy. This section is essentially a review of [21] and companion works [26, 41]. We develop on their formulas for black hole entropy and provide with corrections, which follow mainly from the stringy exclusion principle. The final result for the degeneracy of one-quarter BPS dyons in compactifications can be shown to agree with the microscopic formula (2.11).

The idea of [21] is to compute the contribution to the elliptic genus coming from the fields on , which are dual to the chiral primary states of the CFT. The elliptic genus is nevertheless formulated in the R sector, and thus to count primary states, we must first do a spectral flow transformation to the NS sector. This map consists on the identification

(3.1)
(3.2)
(3.3)

where the subscripts denote the R and NS sectors, and are the Virasoro and R-symmetry generators respectively, and are the left and right central charges. Under this transformation, polar states essentially map to chiral primary states, which we can count from the spectrum of Kaluza-Klein fields on [20, 35, 36]. These include not only the contribution coming from the supergravity fields but also the contribution of M2 and anti-M2 branes wrapping holomorphic two-cycles on the Calabi-Yau. The black hole partition function is obtained by performing a modular transformation, after flowing to the R sector.

The main result of [21] is the factorization of the partition function (index), over the chiral primary states, as the square of the topological string partition function . Essentially the result is

(3.4)

where the trace only goes through the chiral primary states (ch.p.); for simplicity we have omitted a factor in . In the NS sector, chiral primary states obey the condition , which maps to the condition in the R sector (3.1). In addition, chiral primary states can carry charges under currents . In the dilute gas approximation of [21], the trace in the bulk is a trace over BPS multi-particle states [36, 35, 20] with arbitrary spin and occupation numbers. As a consequence, the result of the trace is a formal infinite product over all the quantum numbers, which can be related to using the Gopakumar-Vafa invariants [6, 7]. That is,

(3.5)

This is the key step that allows the authors in [21] to make a connection with the OSV conjecture [5]. Here is the magnetic flux on the sphere , induced by the M5-brane wrapping a divisor four cycle in the homology class Poincare dual to with .

The factors and arise essentially from the contributions of respectively M2-branes and anti-M2-branes wrapping holomorphic cycles in the Calabi-Yau; there is also a contribution coming from the supergravity fields but they will not be relevant for the discussion of compactifications, which is our main interest in this section. What allows the sum over arbitrary M2 and anti-M2 charges is the fact that in AdS space, a brane and its anti-brane can preserve mutual supersymmetries. Indeed, a M2-brane wrapping a holomorphic cycle , siting at the origin of and at the north pole of preserves the same set of supersymmetries as an anti-M2-brane wrapping the same cycle , sitting at the origin of but now at the south pole of [41, 21]. The fact that these are supersymmetric configurations on will play a very important role in the remaining of the letter.

If the theory has supersymmetry, for example when , the partition function (3.5) simplifies considerably, that is,

(3.6)

Here parametrizes the class of , which is Poincare dual to . The coefficient in the product is the Euler character of , which allows for a generalization to other compactifications. In the case of supersymmetry this partition function is trivially one.

Formula (3.4) is valid only in the limit of very large central charge and for low density of chiral primaries, and so it is not the complete answer. The reason is that it does not take into account the stringy exclusion principle, which puts a bound on the total spin of the multi-particle states, that is,

(3.7)

It makes sense as a grand canonical partition function valid for infinite central charge, in which case the stringy exclusion principle constraint can be relaxed. The exclusion principle can not be seen in perturbation theory on , because from the bulk point of view the multi-particle states are free bosonic excitations with no limit in their particle number. Instead, for finite central charge the contribution coming from the perturbative spectrum of Kaluza-Klein fields on must be truncated due to the stringy exclusion principle. Since , by supersymmetry, then the bound on imposes a bound on . Moreover, we have for a static solution and so is also bounded. Therefore, only a finite number of states contribute to (3.4).

Physically, adding M2-branes sitting at the north pole adds non-zero angular momentum

(3.8)

much like an electron in a background magnetic field, while the anti-M2-branes, because they sit at the south pole, contribute with the same sign angular momentum, that is, . Therefore, flowing to the R sector, we find that the state carries R-charge

(3.9)

We then see that the exclusion principle gives a bound on the number of M2 and anti-M2 branes.

The trace in the R sector must contain only states that do not form black holes, up to a spectral flow transformation. In terms of the Virasoro charges this implies

(3.10)

in the R sector, where we have reincorporated a factor. Here is the total M2-brane charge where and are respectively the M2 and anti-M2 charges, and , with the intersection matrix of the Calabi-Yau. Since and is not unimodular, lives on the lattice with the lattice and its dual under the metric ; the quotient removes spectral flow charges [31]. Since the configuration is static, that is, in the NS sector, the condition becomes

(3.11)

where we used the fact that , with and [42]. We can show that (3) is spectral flow invariant. In particular, for or CHL orbifolds this condition becomes

(3.12)

with for the respectively. Here we have used the fact that the only non-vanishing components of are and permutations, together with and . Setting , and we obtain precisely the effective central charges (2.13).

The formula (3.4) also misses important degeneracy factors when taking the trace over the chiral primaries. In the limit when the M2-brane charge is parametrically smaller than , which we are taking to be large, these degeneracy factors are irrelevant for the purpose of arriving at (3.4). This is the dilute gas approximation of [21]. However, since our main interest is for finite central charge, we need to take into account those degeneracy factors. Essentially we follow the discussion in [26]. Under spectral flow from NS to R sector the chiral primaries, which are annihilated by , flow to lowest weight states because flows to . For example the R vacuum corresponds to a lowest weight state with with the level. Therefore acting with we generate the full multiplet, which leads to a degeneracy of states. In addition, these states have to be tensored with the zero modes of the centre of mass multiplet10 which carry spin . The total angular momentum after including the contribution of the M2-branes is

(3.13)

which leads to a degeneracy

(3.14)

Substituting in this expression the values of and for the and CHL examples, that is, and , one obtains precisely

(3.15)

which we identify with the measure factor in the first line of expression (2.11). Since degeneracy is always positive we must have

(3.16)

which is the bound implied by the stringy exclusion principle [20]. In the limit when this bound can be relaxed which is why one obtains the infinite products (3.5).

In addition to the degeneracy, we need to tensor with the states associated with the quantization of fluctuations of the M2 and anti-M2-branes wrapping holomorphic cycles in the Calabi-Yau. For they can be identified with the degeneracies of M2-branes and anti-M2-branes wrapping , which are given by the dedekind function (3.6). These explain the factors in the second line of (2.11).

Assembling all the factors, we construct, in the R sector, the polar partition function

(3.17)

where the sum is over the states obeying the condition (3.12) and (3.16). The black hole partition function is obtained after a modular transformation [21], that is,

(3.18)

There are further corrections to this formula coming from other elements of ; they give contributions of the orbifold type (2.6). The parameter is the weight of the elliptic genus under modular transformations and can be determined as follows. We decompose the elliptic genus in spectral flow sectors as , with a multidimensional theta function 11. The function contains the information about black hole degeneracy, while the theta functions contains the states related by spectral flow symmetry. On one hand, from the Siegel modular form (2.7), of weight , one finds Jacobi forms of weight and a single chemical potential . This implies that part of the Jacobi form that contains the information about black hole degeneracy, which is a vector valued modular form, must have weight , and hence also . On the other hand, since has weight with the second Betti number of the Calabi-Yau, we find that the weight of the elliptic genus is . For the compactification we have and thus . Similarly for the other CHL compactifications we have [40], which also gives !

The black hole degeneracy is computed by an inverse Fourier transform, that is,

(3.20)

with the additional constraint that the sum obeys (3). Specializing the various parameters to the examples and performing the various gaussian integrals in , we obtain almost precisely the one-quarter BPS degeneracy described in section §2.1. The only difference is the contour. While in the formula above we take over the Fourier contour , in the Rademacher expansion one has running over . It looks puzzling how to go from one contour to the other without picking additional contributions. Nevertheless, for the purpose of computing saddle point corrections, both integrals are equally valid. As we explain later, one of the great advantages for using localization is that it naturally picks the Rademacher contour, which then acquires a physical interpretation.

4 Black hole bound states and horizonless geometries

In this section, instead of thinking in terms of M2 and anti-M2 branes wrapping cycles on the Calabi-Yau, we consider an equivalent description in type IIA string theory consisting of a and a configuration wrapping the Calabi-Yau, and carrying fluxes in their worldvolume. This section is essentially a review of the polar configurations of Denef and Moore [27] and their decoupling limit [30]. The main goal is to find a microscopic description for the family of saddle geometries that we propose. Under certain assumptions, we argue that the quantum entropy path integral should be seen as an M-theory path integral with eleven dimensional instanton solutions. Then we propose an effective five dimensional description, which is amenable for using localization.

For the charge configuration of interest, the total charge is zero but the presence of fluxes induce lower dimensional charges due to the couplings of the worldvolume fields to the Ramond-Ramond gauge fields , such as

(4.1)

which generate and charges respectively. Uplifting to M-theory, the pair becomes a Taub-Nut and anti-Taub-Nut configuration, while the and charges lift to and charges and the charges become momentum along the M-theory circle.

From the M-theory point of view the fluxes on the -brane lift to four fluxes [43] in M-theory, with the three-form that couples to M2-branes, that is,

(4.2)

is the self-dual normalizable two form of the Taub-Nut geometry, and is the total flux in the ; and similarly for the brane. Therefore fluxes on the branes map to fluxes in the bulk M-theory.

To be consistent with the brane picture of the previous section, we want to turn on fluxes that generate arbitrary charges as well as charges, but keep fixed the charge, which lifts to the